A154335 A triangular sequence of coefficients of polynomials: p(x,n) = (2*(x - 1)^n * (Sum_{k>=0} (((-1)^n*(2*k + 1)^(n - 1)))*x^k) - (x - 1)^(n + 1)*(Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x).

1, 1, 1, 1, 8, 1, 1, 35, 35, 1, 1, 126, 394, 126, 1, 1, 417, 3062, 3062, 417, 1, 1, 1324, 19895, 44680, 19895, 1324, 1, 1, 4111, 117021, 503827, 503827, 117021, 4111, 1, 1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1, 1, 38333, 3464840, 42960752, 137516234, 137516234, 42960752, 3464840, 38333, 1

Offset

0,5

Comments

Row sums are: {1, 2, 10, 72, 648, 6960, 87120, 1249920, 20280960, 367960320, ...}.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1274

Formula

p(x,n) = (2*(x - 1)^n * (Sum_{k>=0} (((-1)^n*(2*k + 1)^(n - 1)))*x^k) - (x - 1)^(n + 1)*(Sum_{k>=0} ((-1)^(n + 1)*k^n)*x^k)/x).

Functional form:

p(x,n) = (2*(-1)^n* 2^(-1 + n)* (-1 + x)^n* LerchPhi(x, 1 - n, 1/2) - (-1)^(1 + n) *(-1 + x)^(1 + n)* PolyLog(-n, x)/x).

t(n,m) = Coefficients(p(x,n)).

Example

{1},
{1, 1},
{1, 8, 1},
{1, 35, 35, 1},
{1, 126, 394, 126, 1},
{1, 417, 3062, 3062, 417, 1},
{1, 1324, 19895, 44680, 19895, 1324, 1},
{1, 4111, 117021, 503827, 503827, 117021, 4111, 1},
{1, 12602, 648616, 4882342, 9193838, 4882342, 648616, 12602, 1},
{1, 38333, 3464840, 42960752, 137516234, 137516234, 42960752, 3464840, 38333, 1}

Mathematica

Clear[p, x, n]; p[x_, n_] = (2*(x - 1)^(n)*Sum[(((-1)^(n)*(2*k + 1)^(n - 1)))*x^k, {k, 0, Infinity}] - (x - 1)^(n + 1)*Sum[((-1)^(n + 1)*k^n)*x^k, {k, 0, Infinity}]/x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

Crossrefs

Sequence in context: A185412 A157148 A220595 * A142467 A142175 A142597

Adjacent sequences: A154332 A154333 A154334 * A154336 A154337 A154338

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 07 2009

Status

approved